RF-impairment calibration has been thought to be the major challenge in enabling reciprocity-based MU-MIMO. One internal self-calibration method, i.e., the first calibration method which does not require over the air feedback and can enable reciprocity-based MU-MIMO based on moderate and large arrays has been demonstrated.
Conventional downlink MU-MIMO schemes have been at the forefront of investigations in the past decade. These schemes promise spectral efficiency increases by using multiple antennas at the base-station and serving multiple users simultaneously without the need for multiple antennas at the user terminals. This is achieved by using knowledge of the channel state information (CSI) between each user and the transmitting base-station. Having CSIT (CSI available at the transmitter) allows the transmitter to precode the user-terminal streams so that each user terminal sees only its own stream. Given a base station with M transmit antennas, K single-antenna user terminals can be served simultaneously, giving roughly a multiplexing gain equal to min(M,K) with respect to a system serving a single terminal.
For the transmitter to achieve this operation reliably, the transmitter needs to have sufficiently accurate CSIT, i.e., the transmitter needs to know the channels between itself and each of the users sufficiently accurately. The techniques used for acquiring CSIT fall into two categories. The first category employs M pilots (one per base-station transmit antenna) in the downlink to allow each user terminal to estimate the channel coefficients between the user-terminal's own antenna(s) and those of the base-station. This operation provides each CSI at each receiving user-terminal (CSIR) regarding the channel between each base-station transmit antenna and the user-terminal receive antennas. The CSIR, i.e., the CSI information available at each user-terminal, is then fed back to the transmitter by use of uplink transmissions to provide CSIT, i.e., CSI at the transmitting base-station. This category of CSIT acquisition schemes have two overheads: (i) a downlink pilot overhead, which scales linearly with M (then number of antenna elements at the transmitting base-station); (b) an uplink feedback overhead, responsible for making available to the base-station the channels between each user-terminal and each base-station antenna. In the case, each user terminal has a single antenna, and the uplink feedback is responsible for providing to the base-station the MK channel coefficients (complex-scalar numbers), one coefficient for each channel between each user terminal antenna and each base-station antenna. Although the uplink overhead could in principle be made to grow linearly with min(K,M), with the methods used in practice, this overhead grows as the product of M and K. The downlink overhead limits the size of the antenna array, M, that can be deployed. Similarly, the uplink overheads limit both M and K, as the overheads grow very fast with respect to increasing M and K.
The second category of CSIT acquisition techniques is referred to herein as reciprocity-based training schemes. They exploit a property of the physical wireless channel, known as channel reciprocity, to enable, under certain suitably chosen (M,K) pairs, very high-rate transmission with very efficient CSIT training. In particular, pilots are transmitted in the uplink by each user (K pilots are needed, but more could be used) and the corresponding pilot observations at the base-station are directly used at to form the precoder for downlink transmission. If the uplink training and the following downlink data transmission happen close enough in time and frequency (within the coherence time and the coherence bandwidth of the channel) then, the uplink training provides directly the required (downlink channel) CSI at the transmitter, since the uplink and the downlink channels at the same time and frequency are the same. In this category of techniques, the uplink overheads scale linearly with K, i.e., with the number of user terminals that will be served simultaneously. These schemes are also typically envisioned as relying on TDD (Time Division Duplex) in order to allow uplink training and downlink transmission within the coherence bandwidth of the user terminal channel with a single transceiver shared for uplink and downlink data transmission.
One attractive aspect of reciprocity-based training schemes is that one can keep on increasing the size of the transmit antenna array, M, making it “massive”, without incurring any increase in the training overheads. Although with M>K, increasing M does not increase the number of simultaneously multiplexed streams, K, (i.e., K streams are simultaneously transmitted, one to each user), increasing M induces significant “beamforming” gains on each stream (which translate to higher rate per stream), at no additional cost in training. Alternatively, increasing M allows reducing the transmit power required to yield a target rate to a user terminal, thereby allowing for greener transmission schemes.
The challenge with reciprocity based training schemes is that the “compound” uplink and downlink channels at the same time and frequency are not the same. Specifically, although the uplink and downlink physical channel components are the same, each compound channel between a “source node” (responsible for transmitting an information-bearing signal from the transmit antenna) and a destination node (attached to the receive antenna) includes additional impairments due to the transmitter (the circuitry, at the transmitter) and the receiver (the circuitry, at the transmitter). When the transmitter and receiver roles are interchanged, different impairments occur at each node, thereby rendering the two compound channels non-reciprocal.
However, these transmitter/receiver impairments vary or drift, very slowly with time (seconds to minutes to longer, depending on the quality of the circuitry). As a result, this opens the door for infrequent calibration, as a method to compensate for these transmitter/receiver impairments and bring reciprocity based MU-MIMO to fruition.
Reciprocity-Based Massive MU-MIMO
Consider the problem of enabling MU-MIMO transmission from an array of M transmit antennas to K single-antenna user terminals. The downlink (DL) channel between the i-th base-station transmitting antenna and the j-th user terminal is given by{right arrow over (yji)}={right arrow over (rj)}{right arrow over (hji)}{right arrow over (ti)}{right arrow over (xi)}+{right arrow over (zji)}where {right arrow over (xi)}, {right arrow over (hji)}, {right arrow over (yji)}, {right arrow over (zji)}, denote the transmitted signal from base-station antenna i, the DL channel between the two antennas, the observation and noise at the receiver of user terminal j, respectively. The scalar (complex) coefficient {right arrow over (rj)} contains the amplitude and phase shifts introduced by RF-to-baseband conversion hardware (gain control, filters, mixers, A/D, etc.) at the receiver of user terminal j. Similarly, the scalar (complex) coefficient {right arrow over (ti)} contains the amplitude and phase shifts introduced by the baseband-to-RF conversion hardware (amplifiers filters, mixers, A/D, etc.) at the transmitter generating the signal to be transmitted by base-station antenna i.
Similarly the uplink channel between the j-th user terminal and the i-th base-station antenna is given by
            y      ij        ←    =                              r          i                ←            ⁢                        h          ij                ←            ⁢                        t          j                ←            ⁢                        x          j                ←              +                  z        ij            ←      where
            x      j        ←    ,            h      ij        ←    ,            y      ij        ←    ,            z      ij        ←    ,denote the transmitted signal from user terminal j, the uplink (UL) channel between the two antennas, the observation and noise at the receiver of base-station antenna i, respectively. The scalar (complex) coefficient  contains the amplitude and phase shifts introduced by RF-to-baseband conversion hardware (gain control, filters, mixers, A/D, etc.) at the receiver of base-station antenna i. Similarly, the scalar (complex) coefficient  contains the amplitude and phase shifts introduced by the baseband-to-RF conversion hardware (amplifiers filters, mixers, A/D, etc.) at the transmitter generating the signal to be transmitted by user terminal j.
In the uplink, we have the following model:
      y    ←    =                    R        ←            ⁢              H        ←            ⁢              T        ←            ⁢              x        ←              +          z      ←      where  is the vector of dimension K×1 (i.e., K rows by l column) comprising the user symbols on subcarrier n at symbol time t,  is the M×K channel matrix that includes the constant carrier phase shifts and the frequency-dependent constant in time phase shifts due to the relative delays between the timing references of the different terminals,  and  are the received signal vector and noise at the user terminals,
      R    ←    =                    diag        ⁡                  (                                                    r                1                            ←                        ,                                          r                2                            ←                        ,            …            ⁢                                                  ,                                          r                M                            ←                                )                    ⁢                          ⁢      and      ⁢                          ⁢              T        ←              =                  diag        ⁡                  (                                                    t                1                            ←                        ,                                          t                2                            ←                        ,            …            ⁢                                                  ,                                          t                K                            ←                                )                    .      
In the downlink, we have the following model{right arrow over (y)}={right arrow over (x)}{right arrow over (T)}{right arrow over (H)}{right arrow over (R)}+{right arrow over (z)}where {right arrow over (x)} is the (row) vector of user symbols on subcarrier n at symbol time t, {right arrow over (H)} is the K×M channel matrix that includes the constant carrier phase shifts and the frequency-dependent constant in time phase shifts due to the relative delays between the timing references of the different terminals, {right arrow over (y)} and {right arrow over (z)} are the received signal (row) vector and noise at the user terminals, {right arrow over (R)}=diag({right arrow over (r1)}, {right arrow over (r2)}, . . . , {right arrow over (rK)}) and {right arrow over (T)}=diag({right arrow over (t1)}, {right arrow over (t2)}, . . . , {right arrow over (tM)})
The matrices ,  {right arrow over (R)} and {right arrow over (T)} are unknown locally constant diagonal matrices. For purposed herein, “locally constant” means that they might vary over very long time (certainly, much longer than the coherence time of the channel), mainly due to thermal drift effects, but they do not depend on any “fast effects” such as frequency offsets, and propagation time-varying fading, since these effects are all already taken care of by the timing and carrier phase synchronization, and included in the matrices {right arrow over (H)} and . By reciprocity of the physical channel, the following holds{right arrow over (H)}=For simplicity, the thermal noise is neglected. In order to estimate the downlink channel matrix, the K user terminals send a block of K OFDM symbols, such that the uplink-training phase can be written as
            Y      tr        ←    =                    R        ←            ⁢              H        ←            ⁢              T        ←            ⁢                        X          tr                ←              +    noise  where  is a scaled unitary matrix. Hence, the base-station can obtain the channel matrix estimate
                    Y        tr            ←        ⁢                  X        tr            ←        =                    R        ←            ⁢              H        ←            ⁢              T        ←              +    noise  However, in order to perform downlink beamforming we need the downlink matrix {right arrow over (T)}{right arrow over (H)}{right arrow over (R)}. While reciprocity ensures that the physical channel component in the uplink estimated channel yields immediately the corresponding component in the downlink channel (it is assumed that uplink training and downlink data transmission occur in the same channel coherence time), the transmit and receive diagonal matrices need to be known for the downlink, while the product of those matrices for the uplink and the channel matrix {right arrow over (H)}= are known, which are generally arbitrarily related.Prior Art on Relative Calibration
As a prelude to describing one relative calibration method, notice that the downlink channel matrix {right arrow over (T)}{right arrow over (H)}{right arrow over (R)} is not entirely needed to perform beamforming. In fact, only the column-space of this matrix is needed; that is, any matrix formed by {right arrow over (T)}{right arrow over (H)}A, where A is some arbitrary invertible constant diagonal matrix, is good enough for any kind of beamforming. For example, consider Zero Forced Beamforming (ZFBF). The ZFBF precoding matrix can be calculated asW=Λ1/2[AH{right arrow over (H)}H{right arrow over (T)}H{right arrow over (T)}{right arrow over (H)}A]−1AH{right arrow over (H)}H{right arrow over (T)}H where Λ is a diagonal matrix that imposes on each row of the matrix W, the row normalization ∥wm∥2=1, for all m. Hence, the ZFBF precoded signal in the downlink will be
                              y          →                =                ⁢                                            u              →                        ⁢            W            ⁢                          T              →                        ⁢                          H              →                        ⁢                          R              →                                +                      z            →                                                  =                ⁢                                            u              →                        ⁢                                                            Λ                                      1                    /                    2                                                  ⁡                                  [                                      A                    ⁢                                                                  H                        →                                                                  H                                                                      H                                        ⁢                                                                  T                        →                                            H                                        ⁢                                          T                      →                                        ⁢                                          H                      →                                        ⁢                    A                                    ]                                                            -                1                                      ⁢            A            ⁢                                          H                →                                          H                                              H                        ⁢                                          T                →                            H                        ⁢                          T              →                        ⁢                          H              →                        ⁢                          R              →                                +                      z            →                                                  =                ⁢                                            u              →                        ⁢                          Λ                              1                /                2                                      ⁢            A            ⁢                                                                             -                  1                                            ⁢                              R                →                                              +                      z            →                              
We notice that the resulting channel matrix is diagonal, provided that K≤M. It follows that the problem is how to estimate {right arrow over (T)}{right arrow over (H)} up to the left multiplication by some known matrix A, from the uplink training observation  knowing that {right arrow over (H)}=. Following the relative calibration procedure of Shepard et al. in “Argos: Practical Many-Antenna Base Stations” (Mobicom 2012, pp. 53-64; hereinafter “Argos”), the fact that the diagonal matrices , , {right arrow over (R)}, and {right arrow over (T)} are essentially constant in time for intervals much longer than the slot duration can be exploited (the calibration procedure may be repeated periodically, every some tens of seconds or even more, depending on the hardware stability, temperature changes, etc.).
The procedure, amounting to one calibration method, consists of the following steps:                Training from a calibration-reference base-station antenna, e.g., antenna 1: sending a pilot symbol from base-station antenna 1 to all other base-station antennas, i.e., to the set of base-station antennas S={2, 3, . . . , M}. The received signal at the BS antennas, S, is given by        
      y          S      ←      1        =                              R          S                ←            ⁢              h                  S          ←          1                    ⁢                        t          1                →              +                  z        ←                    S        ←        1                            where {right arrow over (t1)} is the coefficient due to base-station reference antenna (i.e., antenna 1) transmit RF chain,        
                    R        S            ←        =          diag      ⁡              (                                            r              2                        ←                    ,                                    r              3                        ←                    ,          …          ⁢                                          ,                                    r              M                        ←                          )              ,i.e., it is a diagonal matrix containing the coefficients due to the other base-station antennas receive RF chains, the (M−1)×1 vector hS←1 denotes physical channel from reference base-station antenna 1 to the rest of the base-station antennas, and the (M−1)×1 vector  represents thermal noise at the (M−1) non-transmitting base-station antennas.                Training from the base-station antennas in the set S to the calibration-reference antenna 1: the base-station antennas 2, 3, . . . , M respond with a sequence of M−1 symbols each, to form a (proportional to) unitary training matrix (one special case corresponds to sending one pilot each at a time). The signal received by the calibration-reference antenna is given by        
      y          S      →      1        =                              X          →                calib            ⁢                        T          S                →            ⁢              h                  S          →          1                    ⁢                        r          1                ←              +                  z        ←                    S        →        1            where  is the coefficient due to the calibration-reference antenna receive RF chain.                Calibration process: premultiplying by {right arrow over (X)}calibH the BS obtains        
                    X        →            calib      H        ⁢          y              S        →        1              =                              T          S                →            ⁢              h                  S          →          1                    ⁢                        r          1                ←              +    noise                  Now, notice that, due to physical channel reciprocity, we have hS→1=hS←1. Hence, for each m=2, 3, . . . , M, the base station can compute the ratios        
                                                        [                                                                    X                    →                                    calib                  H                                ⁢                                  y                                      s                    →                    1                                                              ]                                      m              -              1                                                          [                              y                                  s                  ←                  1                                            ]                                      m              -              1                                      =                ⁢                                                                                                                        t                      m                                        →                                    ⁡                                      [                                          h                                              s                        →                        1                                                              ]                                                                    m                  -                  1                                            ⁢                                                r                  ←                                1                                      +            noise                                                                                                                          r                      m                                        ←                                    ⁡                                      [                                          h                                              s                        ←                        1                                                              ]                                                                    m                  -                  1                                            ⁢                                                t                  →                                1                                      +            noise                                                  =                ⁢                                                                              t                  m                                →                                                              r                  m                                ←                                      ⁢                                                            r                  1                                ←                                                              t                  1                                →                                              +          noise                                    At the end of the calibration process, for sufficiently high SNR such that the noise can be neglected, one has obtained the diagonal calibration matrix  where        
      α    1    =                    r        1            ←        /                  t        1            →      is an irrelevant constant term that depends only on the calibration-reference antenna up and down modulation chains.
At this point, the desired downlink channel matrix can be obtained from the calibration matrix  and the uplink estimated channel matrix  simply by multiplication with the uplink estimated channels. In particular, we have,
            T      →        ⁢                  R        ←                    -        1              ⁢          α      1                                                              α              1                        ⁢                          T              →                        ⁢                                          R                ←                                            -                1                                      ⁢                                          Y                ←                            tr                        ⁢                                          X                ←                            tr              H                                =                    ⁢                                                    α                1                            ⁢                              T                →                            ⁢                                                R                  ←                                                  -                  1                                            ⁢                              R                ←                            ⁢                              H                ←                            ⁢                              T                ←                                      +            noise                                                        =                    ⁢                                                    T                →                            ⁢                                                H                  →                                ⁡                                  [                                                            α                      1                                        ⁢                                          T                      ←                                                        ]                                                      +            noise                                                        =                    ⁢                                                    T                →                            ⁢                              H                →                            ⁢              A                        +            noise                              where A=α1
The self-calibration process of this calibration method takes at least M OFDM symbols, one symbol for the pilot from reference antenna to all other base-station antennas, and M−1 OFDM symbols to send orthogonal training sequences from all the other base-station antennas to the calibration-reference antenna.
This calibration method has its limitations. First, note that the relative calibration of each base-station antenna with respect to the reference antenna is formed as the ratio of two observations, and, in particular, by dividing
            [                                    X            →                    calib          H                ⁢                  y                      s            →            1                              ]              m      -      1                  with      ⁢                          [              y                  s          →          1                    ]              m      -      1.      The noise in the dividing term [yS→1]m-1, can cause a large estimation error in the calibration estimate. This effect was indeed noticed by the developers of this calibration method: “Another challenge we encountered while performing our indirect calibration approach is the significant amplitude variation for the channels between the reference antenna 1 and other antennas. This is due to the grid-like configuration of our antenna array where different pairs of antennas can have very different antenna spacings. According to our measurement, the SNR difference can be as high as 40 dB, leading to a dilemma for us to properly choose the transmission power for the reference signal.” Their solution was to carefully place the reference antenna with respect to the rest of the base-station antennas: “we isolate the reference antenna from the others, and place it in a position so that its horizontal distance to the other antennas are approximately identical. Such placement of the reference antenna does not affect the calibration performance due to our calibration procedure's isolation of the radio hardware channel from the physical channel.”
Such a need for careful placement of the reference antenna with respect to the rest of the transmitting antennas is a significant limiting factor in deployments relying on this calibration method, which significantly limits their efficacy in downlink MU-MIMO deployments from sets of non-collocated antennas.
In general, for noise robustness purposes, much larger blocks and maximal ratio combining of the received power can be used, such that we sent D pilot symbols from reference antenna 1 to the other base-station antennas, and M−1 orthogonal training sequences over (M−1)D symbols from the other base-station antennas to the reference one, achieving a factor D in signal to noise ratio for calibration, where D≥1 is some sufficiently large integer in order to improve performance. However, this does not eliminate the inherent limitations of the Argos calibration methods especially for scalable and distributive deployments.